GRE word problem 1

Question

Question: Scott starts jogging from point $X$ to point $Y$. A half-hour later his friend Garrett who jogs $1$ mile per hour slower than twice Scott's rate starts from the same point and follows the same path. If Gareth overtakes Scott in $2$ hours, how many miles will Garrett have covered?

Answer: $\frac{10}{3}$ miles. Why?

Answer 1

Let $x$ = Scott's speed, then Garrett's speed is $2x-1$.

For the first half hour, Scott covered $0.5x$ miles, and for the next $2$ hours, Scott covers $2x$ miles, and Garrett covered $2(2x-1)$ miles. So:

$0.5x + 2x = 2(2x-1)$.

Thus: $x = \dfrac{4}{3}$.

Thus Scott covers a total distance of: $2.5\cdot \dfrac{4}{3} = \dfrac{10}{3}$ miles, which is the same distance that Garrett traveled.

Answer 2

A start: The standard way would be to let Scott's speed be $s$. Then Gareth's speed is $2s-1$. Note that $2.5s=2(2s-1)$. Our heroes are slow.

Answer 3

Let Scott's speed be $x$, then we know that Garrett's speed is $2x-1$.
Distance covered by Scott after $30$mins is $\frac{x}{2}$, and since he took $2$ hours to catch up, we have $$\frac{x}{2}=2(x-1).$$ Solve for $x$ to get (mouse over for answer, don't cheat!)

$x=4/3$

Now, compute $2x+x/2$ to get

$10/3$